30
VIII – Cauchy Theory
There are also close links between these theories and the problem of approximating holomorphic functions on a given domain G by simple polynomial functions or, when this is not possible, by rational functions without
poles in G. If for example G is simply connected, and only in this case, any
holomorphic function f on G is the limit of a sequence of polynomials in z
converging uniformly to f on every compact subset of G. This result and
more general ones are due to Carl Runge
22 (1885).
22 In fact, Runge did not see that the approximation by rational functions led to
the result in question in the case of a simply connected domain. On another note,
Runge was interested in atomic spectra in the hope of finding simple formulas
that would allow their frequencies to be computed, as had already been done by
Balmer for the hydrogen atom. We now know that this amounts to calculating
the corresponding eigenvalues of the Schr¨ odinger operator. This problem is still
too hard, the helium atom, and all the more the following ones, continuing to
resist all exact solutions. After 1900, when Felix Klein created the first applied
mathematics team in G¨ ottingen, one of his recruits would be Runge, the first
leading expert of numerical analysis. He also recruited Ludwig Prandtl, who
remained until 1945 the greatest German expert of aerodynamics, perhaps even
the greatest world expert,while Prandtl’s first brilliant student, the Hungarian
Theodor von K´ arm´ an who emigrated to CalTech at the end of the 1920s, would
play the same role in the USA until the end of the 1950s. See Paul A. Hanle,
Bringing Aerodynamics to America (MIT Press, 1982).
VIII – Cauchy Theory
There are also close links between these theories and the problem of approximating holomorphic functions on a given domain G by simple polynomial functions or, when this is not possible, by rational functions without
poles in G. If for example G is simply connected, and only in this case, any
holomorphic function f on G is the limit of a sequence of polynomials in z
converging uniformly to f on every compact subset of G. This result and
more general ones are due to Carl Runge
22 (1885).
22 In fact, Runge did not see that the approximation by rational functions led to
the result in question in the case of a simply connected domain. On another note,
Runge was interested in atomic spectra in the hope of finding simple formulas
that would allow their frequencies to be computed, as had already been done by
Balmer for the hydrogen atom. We now know that this amounts to calculating
the corresponding eigenvalues of the Schr¨ odinger operator. This problem is still
too hard, the helium atom, and all the more the following ones, continuing to
resist all exact solutions. After 1900, when Felix Klein created the first applied
mathematics team in G¨ ottingen, one of his recruits would be Runge, the first
leading expert of numerical analysis. He also recruited Ludwig Prandtl, who
remained until 1945 the greatest German expert of aerodynamics, perhaps even
the greatest world expert,while Prandtl’s first brilliant student, the Hungarian
Theodor von K´ arm´ an who emigrated to CalTech at the end of the 1920s, would
play the same role in the USA until the end of the 1950s. See Paul A. Hanle,
Bringing Aerodynamics to America (MIT Press, 1982).